坎宁安函数

Cunningham function Maple animation

坎宁安函数又称为皮尔逊-坎宁安函数（Pearson-Cunningham function)是英国数学家坎宁安在1908年首先研究的特殊函数,，定义如下：

${\omega }_{m,n}(x)=\frac{e^{-x+\pi i(m/2-n)}}{\Gamma (1+n-m/2)}U(m/2-n,1+m,x).$

其中U为特里科米函数。

坎宁安在是在用多變數擴展的埃奇沃斯級數，依機率密度函數的矩來近似機率密度函數時用到坎宁安函数，坎宁安函数和一維或多維常係數的擴散方程有關

坎宁安函数是下列微分方程的解

$x{X}^{″}+(x+1+m)X^{\prime }+(n+\frac{1}{2}m+1)X.$

与其他函数的关系

• ${\omega }_{m,n}(x)=\frac{exp(-x+(1/2\ast I)\ast \pi \ast m-I\ast \pi \ast n)\ast \Gamma (m)\ast HeunB(-2\ast m,0,2+4\ast n,0,\sqrt{(}x))}{\Gamma (1+n-(1/2)\ast m)\ast x^m\ast \Gamma ((1/2)\ast m-n)}$

$+\frac{exp(-x+(1/2\ast I)\ast Pi\ast m-I\ast \pi \ast n)\ast \Gamma (-m)\ast HeunB(2\ast m,0,2+4\ast n,0,\sqrt{(}x))}{\Gamma (1+n-(1/2)\ast m)\ast \Gamma (-(1/2)\ast m-n)}$

• ${\omega }_{m,n}=\frac{WhittakerM(0,-1/2,-x+I\ast \pi \ast ((1/2)\ast m-n))\ast exp(-(1/2)\ast x+(1/2\ast I)\ast \pi \ast ((1/2)\ast m-n))\ast WhittakerW(1/2+n,(1/2)\ast m,x)\ast exp((1/2)\ast x)}{\Gamma (1+n-(1/2)\ast m)\ast x^{(1/2+(1/2)\ast m)}}$

级数展开

${\omega }_{0.5,0.5}(x)=(1/80640)\ast (120960\ast \sqrt{(}2)\ast \Gamma (3/4{)}^2\ast \sqrt{(}x)-141120\ast \sqrt{(}2)\ast \Gamma (3/4{)}^2\ast x^{(}3/2)+77616\ast \sqrt{(}2)\ast \Gamma (3/4{)}^2\ast x^{(}5/2)-27720\ast \sqrt{(}2)\ast \Gamma (3/4{)}^2\ast x^{(}7/2)+7315\ast \sqrt{(}2)\ast \Gamma (3/4{)}^2\ast x^{(}9/2)+(141120\ast I)\ast \sqrt{(}2)\ast \Gamma (3/4{)}^2\ast x^{(}3/2)+(27720\ast I)\ast \sqrt{(}2)\ast \Gamma (3/4{)}^2\ast x^{(}7/2)-(100800\ast I)\ast \pi \ast x-(7315\ast I)\ast \sqrt{(}2)\ast \Gamma (3/4{)}^2\ast x^{(}9/2)-(77616\ast I)\ast \sqrt{(}2)\ast \Gamma (3/4{)}^2\ast x^{(}5/2)-40320\ast \pi +(75600\ast I)\ast \pi \ast x^2+100800\ast \pi \ast x+(40320\ast I)\ast \pi -75600\ast \pi \ast x^2-(120960\ast I)\ast \sqrt{(}2)\ast \Gamma (3/4{)}^2\ast \sqrt{(}x)+32760\ast \pi \ast x^3-(32760\ast I)\ast \pi \ast x^3-9945\ast \pi \ast x^4+(9945\ast I)\ast \pi \ast x^4+80640\ast {\pi }^{(}3/2)\ast O(x^{(}9/2))\ast \sqrt{(}x))/({\pi }^{(}3/2)\ast \sqrt{(}x))$

参考文献

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